{
  "id": "1804.06400",
  "version": "v1",
  "published": "2018-04-17T17:56:33.000Z",
  "updated": "2018-04-17T17:56:33.000Z",
  "title": "The Eisenstein ideal with squarefree level",
  "authors": [
    "Preston Wake",
    "Carl Wang-Erickson"
  ],
  "comment": "44 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We use pseudodeformation theory to study the analogue of Mazur's Eisenstein ideal with certain squarefree levels. Given a prime number $p>3$ and a squarefree number $N$ satisfying certain conditions, we study the Eisenstein part of the $p$-adic Hecke algebra for $\\Gamma_0(N)$, and show that it is a local complete intersection and isomorphic to a pseudodeformation ring. We also show that in certain cases, the Eisenstein ideal is not principal and that the cuspidal quotient of the Hecke algebra is not Gorenstein. As a corollary, we prove that \"multiplicity one\" fails for the modular Jacobian in these cases. In a particular case, this proves a conjecture of Ribet.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-17T17:56:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "squarefree level",
      "local complete intersection",
      "adic hecke algebra",
      "mazurs eisenstein ideal",
      "cuspidal quotient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}